%0 Journal Article
%T On the Link between Stopping Time and Non-Trivial Cycles in the Collatz Problem
%A Lionel Laurore
%J Advances in Pure Mathematics
%P 351-389
%@ 2160-0384
%D 2025
%I Scientific Research Publishing
%R 10.4236/apm.2025.156018
%X The Collatz Conjecture asserts that for all positive integers <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>s</mi>
</math>

, every Syracuse integer sequence defined by <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>T</mi><mrow><mo>(</mo>
   <mi>s</mi>
  <mo>)</mo></mrow><mo>=</mo><mrow><mi>s</mi><mo>/</mo><mn>2</mn></mrow>
  </mrow>
</math>

 if <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>s</mi>
</math>

 is even, and <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>T</mi><mrow><mo>(</mo>
   <mi>s</mi>
  <mo>)</mo></mrow><mo>=</mo><mrow><mrow>
   <mrow><mo>(</mo>
    <mrow>
     <mn>3</mn><mi>s</mi><mo>+</mo><mn>1</mn></mrow>
   <mo>)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow>
  </mrow>
</math>

 otherwise, eventually reaches 1 after a finite number of iterations. The stopping time of an integer is the smallest number of iterations required for the sequence to fall below its starting value, while the total stopping time measures the iterations needed to reach 1. In this paper, we revisit the notion of stopping time by introducing the coefficient stopping time, defined as the smallest value of n such that the coefficient of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>s</mi>
</math>

 in <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mi>T</mi>
   <mi>n</mi>
  </msup>
  <mrow><mo>(</mo>
   <mi>s</mi>
  <mo>)</mo></mrow></mrow>
</math>

, expressed as <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mrow><mrow>
   <msup>
    <mn>3</mn>
    <mi>r</mi>
   </msup>
   </mrow><mo>/</mo><mrow>
   <msup>
    <mn>2</mn>
    <mi>n</mi>
   </msup>
   </mrow></mrow>
  </mrow>
</math>

, is less than 1. Building on foundational results by Lynn E. Garner (1981), we leverage recent computational results by David Barina to extend Garner&#8217;s estimation regarding the minimal length of non-trivial cycles. Specifically, we demonstrate the non-existence of non-trivial cycles of length <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>n</mi><mo>&#x003C;</mo><mn>19478780533</mn></mrow>
</math>

, thus improving upon the previous result by Shalom Eliahou (2021). We subsequently show that this result can be generalized to all integers <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>n</mi>
</math>

. We also introduce new properties concerning the behavior of Syracuse sequences modulo <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mn>2</mn>
   <mi>n</mi>
  </msup>
  </mrow>
</math>

, which play a central role in our approach.
%K Collatz Problem
%K Stopping Time
%K Coefficient Stopping Time
%K Non-Trivial Cycles
%K Garner&#8217
%K s Main Theorem
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=143316